Question

rays ba and bc are perpendicular. point d lies in the interior of ∠abc. if m∠abd=(3x + 5)° and m∠dbc=(5x - 27)°, find m∠abd and m∠dbc. fill in the blanks below with the correct responses for the value of x, and the two angle measures

Explanation:

Step1: Set up equation

Since $\overrightarrow{BA}$ and $\overrightarrow{BC}$ are perpendicular, $\angle ABC = 90^{\circ}$, and $\angle ABD+\angle DBC=\angle ABC$. So, $(3x + 5)+(5x - 27)=90$.

Step2: Combine like - terms

Combine the $x$ terms and the constant terms on the left - hand side: $3x+5x + 5-27=90$, which simplifies to $8x-22 = 90$.

Step3: Isolate the variable term

Add 22 to both sides of the equation: $8x-22 + 22=90 + 22$, resulting in $8x=112$.

Step4: Solve for x

Divide both sides by 8: $\frac{8x}{8}=\frac{112}{8}$, so $x = 14$.

Step5: Find $\angle ABD$

Substitute $x = 14$ into the expression for $\angle ABD$: $m\angle ABD=(3x + 5)=(3\times14 + 5)=42 + 5=47^{\circ}$.

Step6: Find $\angle DBC$

Substitute $x = 14$ into the expression for $\angle DBC$: $m\angle DBC=(5x - 27)=(5\times14-27)=70 - 27 = 43^{\circ}$.

Answer:

$x = 14$
$m\angle ABD = 47^{\circ}$
$m\angle DBC = 43^{\circ}$